Question 24 of 25intermediate🔍 AnalyzeShort Answer3 marks

How about checking divisibility by 24? Will checking the divisibility by its factors, 4 and 6, work? Why or why not?

Correct Answer

No, checking divisibility by factors 4 and 6 does not work for divisibility by 24.

Reasoning: Factors 4 and 6 share a common factor of 2. LCM(4, 6) = 12, which is not equal to 24. A number divisible by both 4 and 6 is divisible by LCM(4, 6) = 12, but not necessarily by 24.

Counter-example: 12 is divisible by both 4 and 6. But 12 is not divisible by 24.

Correct Method: To check divisibility by 24, check divisibility by 3 and 8. LCM(3, 8) = 24 (since 3 and 8 are coprime).

Exercise: Divisibility Shortcuts for Other Numbers | Q: 2 | (Page Page 18)
For More Understanding

Explanation

The chapter explains that checking divisibility by factors 4 and 6 fails because 4 and 6 share a common factor of 2. LCM(4,6) = 12, not 24. A number divisible by both 4 and 6 is divisible by LCM(4,6) = 12, but not necessarily by 24. For divisibility by 24, we need to check divisibility by 3 and 8, since LCM(3,8) = 24 and 3 and 8 are coprime. Example: 24 is divisible by 3 and 8, and also by 24. Counter-example: 12 is divisible by 4 and 6, but not by 24.

Solution Steps

  1. Identify that factors 4 and 6 share a common factor of 2.

  2. Calculate LCM(4, 6) = 12.

  3. Observe that LCM(4, 6) ≠\neq 24.

  4. Provide counter-example: 12 is divisible by 4 and 6 but not by 24.

  5. Identify correct factors: 3 and 8 (which are coprime).

  6. Calculate LCM(3, 8) = 24.

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